Question

use a half - angle for
$\tan 157.5^{circ}$
$\tan 157.5^{circ}=square$
(simplify your answer, including any radicals. use integers or fractions for a

Explanation:

Step1: Determine the half - angle formula

The half - angle formula for tangent is \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\) or \(\tan\frac{\alpha}{2}=\frac{\sin\alpha}{1+\cos\alpha}\). Let \(\alpha = 315^{\circ}\), then \(\frac{\alpha}{2}=157.5^{\circ}\).

Step2: Find the values of \(\sin\alpha\) and \(\cos\alpha\)

We know that \(\sin315^{\circ}=-\frac{\sqrt{2}}{2}\) and \(\cos315^{\circ}=\frac{\sqrt{2}}{2}\).

Step3: Substitute into the half - angle formula

Using \(\tan\frac{\alpha}{2}=\frac{1 - \cos\alpha}{\sin\alpha}\), substitute \(\alpha = 315^{\circ}\):

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Answer:

\(1-\sqrt{2}\)